O R I G I N A L R E S E A R C H
Fixed point approximation of Picard normal S-iteration process
for generalized nonexpansive mappings in hyperbolic spaces
Mohammad Imdad1•Samir Dashputre2Received: 15 August 2015 / Accepted: 14 July 2016 / Published online: 23 July 2016 Ó The Author(s) 2016. This article is published with open access at Springerlink.com
Abstract In this paper, we establish strong and D-con-vergence theorems for a relatively new iteration process generated by generalized nonexpansive mappings in uni-formly convex hyperbolic spaces. The theorems presented in this paper generalizes corresponding theorems for uni-formly convex normed spaces of Kadioglu and Yildirim (Approximating fixed points of nonexpansive mappings by faster iteration process, arXiv:1402.6530v1 [math.FA],
2014) and CAT(0)-spaces of Abbas et al. (J Inequal Appl 2014:212,2014) and many others in this direction. Keywords Generalized nonexpansive mappings Strong and D-convergence Uniformly convex hyperbolic spaces Picard normal S-iteration process
Introduction
In this paper, N denotes the set of all positive integers while F(T) denotes the set of all fixed points of T, i.e., FðTÞ ¼ fTx ¼ x; x 2 Cg.
Let C be a nonempty subset of normed space X and mapping T : C! C is said to be
(i) nonexpansive, ifkTx Tyk kx yk, for all x; y 2 C,
(ii) quasi-nonexpansive, ifkTx pk kx pk, for all x2 C and p 2 FðTÞ.
Many nonlinear equations are naturally formulated as fixed point problems,
x¼ Tx; ð1:1Þ
where T, the fixed point mapping, may be nonlinear. A solution xof the problem (1.1) is called a fixed point of the mapping T. Consider a fixed point iteration, which is given by
xnþ1¼ Txn;8n 2 N: ð1:2Þ
The iterative method (1.2) is also known as Picard itera-tion or the method of successive substituitera-tion. For the Banach contraction mapping theorem, the Picard iteration converges unique fixed point of T, but it fails to approxi-mate fixed point for nonexpansive mappings, even when the existence of a fixed point of T is guaranteed.
Example 1.1 Consider a self mapping T on [0, 1] defined by Tx¼ 1 x for 0 x 1. Then T is nonexpansive with unique fixed point at x¼1
2. If we choose a starting value x¼ a 6¼1
2, then successive iteration of T yield the sequence f1 a; a; 1 a; . . .g.
Thus, when a fixed point of nonexpansive mappings exists, other approximation techniques are needed to approximate it. In the last fifty years, the numerous num-bers of researchers attracted in these direction and devel-oped iterative process has been investigated to approximate fixed point for not only nonexpansive mapping, but also for some wider class of nonexpansive mappings (see e.g., Agarwal et al. [3], Ishikawa [9], Krasnosel’skiıˇ [12], Mann & Mohammad Imdad
[email protected] Samir Dashputre [email protected]
1 Department of Mathematics, Aligarh Muslim University,
Aligarh Uttar Pradesh, India
2 Department of Applied Mathematics, Shri Shankaracharya
Technical Campus, Shri Shankaracharya Group of Institutions (F.E.T), Junwani, Bhilai 490020, India DOI 10.1007/s40096-016-0187-8
[18], Noor [19], Schaefer [23]), and compare which one is faster.
Sahu [21] has introduced Normal S-iteration Process, whose rate of convergence similar to the Picard iteration process and faster than other fixed point iteration processes (see [21, Theorem 3.6]).
ðNSÞ Normal S-iteration process (see Sahu [21]) defined as follows:
For C a convex subset of normed space X and a non-linear mapping T of C into itself, for each x12 C, the sequencefxng in C is defined by
xnþ1¼ Tyn
yn ¼ ð1 anÞxnþ anTxn; n2 N;
ð1:3Þ wherefang is real sequences in (0, 1).
It brings a following natural question.
Question 1.1 Does there exists an iteration process whose rate of convergence is faster than Normal S-itera-tion process for contracS-itera-tion mappings?
The question have been resolved in affirmative way by Abbas et al. [2], Kadioglu and Yildirim [11, Theorem 5], Thakur et al. [26, Theorem 2.3], developed new iteration processes for approximating the fixed point, as earliest as possible compare Normal S-iteration process. The follow-ing iteration process developed by Kadioglu and Yildirim [11] for approximating the fixed point for nonexpansive mapping and establish some strong and weak convergence theorems in uniformly convex Banach spaces.
ðPNSÞ Picard normal S-iteration process (see Kadioglu and Yildirim [11]) defined as follows: With C, X and T as in (NS), for each x12 C, the sequence fxng in C is defined by
xnþ1¼ Tyn yn ¼ ð1 anÞznþ anTzn zn¼ ð1 bnÞxnþ bnTxn; n2 N; 8 > < > : ð1:4Þ
wherefang and fbng are real sequences in (0, 1).
Remark 1.1 If bn ¼ 0 and an¼ bn ¼ 0 in the process (1.4) then it reduces to Normal S-iteration process (1.3) and Picard iteration process (1.2) respectively.
The purpose of this paper is to establish strong and D-convergence theorems for a new iteration process gener-ated by generalized nonexpansive mappings in uniformly convex hyperbolic spaces. The theorems presented in this paper generalizes corresponding theorems for uniformly convex normed spaces of Kadioglu and Yildirim [11] and CAT(0)-spaces of Abbas et al. [1] and many others in this directions (see Itoh [8], Kim et al. [14], Sahu [21] etc.).
Preliminaries
Let (X, d) be a metric space and C be a nonempty subset of X. Suzuki [24] introduced a class of single valued map-pings called Suzuki-generalized nonexpansive mapmap-pings (or condition (C)), satisfying a condition
1
2dðx; TxÞ dðx; yÞ ¼) dðTx; TyÞ dðx; yÞ;
which is weaker than nonexpansiveness and stronger than quasi nonexpansiveness. The following examples make obvious this fact.
Example 2.1 [24] Define a mapping T on [0, 3] by Tx¼ 0; if x6¼ 3;
1; if x¼ 3:
Then T satisfies condition (C), but T is not nonexpansive.
Example 2.2 [24] Define a mapping T on [0, 3] by Tx¼ 0; if x6¼ 3;
2; if x¼ 3:
Then FðTÞ ¼ f0g 6¼ £ and T is quasi-nonexpansive, but T does not satisfy condition (C).
In [10], Karapinar and Tas introduced some new defi-nitions which are modifications of Suzuki’s-generalized nonexpansive mappings (or condition (C)) as follows. Definition 2.1 Let C be a nonempty subset of a metric space X. The mapping T : C! C is said to be
(i) Suzuki-Ciric mapping (SCC) [10] if 1
2dðTx; TyÞ dðx; yÞ ¼) dðTx; TyÞ Mðx; yÞ where
Mðx; yÞ ¼ maxfdðx; yÞ; dðx; TxÞ; dðy; TyÞ; dðx; TyÞ; dðy; TxÞg
for all x; y2 C;
(ii) Suzuki-KC mapping (SKC) if 1
2dðTx; TyÞ dðx; yÞ ¼) dðTx; TyÞ Nðx; yÞ where
Nðx; yÞ ¼ max
dðx; yÞ;dðx; TxÞ þ dðy; TyÞ
2 ;
dðx; TyÞ þ dðy; TxÞ 2
(iii) Kannan Suzuki mapping (KSC) if 1
2dðTx; TyÞ dðx; yÞ ¼) dðTx; TyÞ dðx; TxÞ þ dðy; TyÞ
2 for all x; y2 C;
(iv) Chatterjea–Suzuki mappings (CSC) if 1
2dðTx; TyÞ dðx; yÞ ¼) dðTx; TyÞ dðy; TxÞ þ dðx; TyÞ
2 for all x; y2 C;
Theorem 2.1 [10] Let T be a mapping on a closed subset C of a metric space X and T satisfy condition SKC. Then dðx; TyÞ 5dðTx; xÞ þ dðx; yÞ holds for x; y 2 C.
Remark 2.1 Theorem2.1holds if one replaces condition SKC by one of the conditions KSC, SCC, and CSC.
Recently, Garcı´a-Falset et al. [7] introduced two gen-eralizations of nonexpansive mappings which in turn include Suzuki generalized nonexpansive mappings con-tained in [24].
Definition 2.2 Let T be a mapping defined on a subset C of metric space X and l 1. Then T is said to satisfy conditionðElÞ, if (for all x; y 2 C)
dðx; TyÞ ldðx; TxÞ þ dðx; yÞ:
Often, T is said to satisfy condition (E) whenever T satisfies conditionðElÞ for some l 1.
Remark 2.2 If T satisfies one of the conditions SKC, KSC, SCC, and CSC, then T satisfies condition El for l¼ 5.
Definition 2.3 Let T be a mapping defined on a subset C of a metric space X and k2 ð0; 1Þ. Then T is said to satisfy the conditionðCkÞ if for all x; y 2 C
kdðx; TxÞ dðx; yÞ ¼) dðTx; TyÞ dðx; yÞ:
For 0\k1\k2\1, the condition ðCk1Þ implies the
conditionðCk2Þ.
The following example shows that the class of mappings satisfying conditions (E) and ðCkÞ for some k 2 ð0; 1Þ is larger than the class of mappings satisfying the condition (C). Example 2.3 [7] For a given k2 ð0; 1Þ, define a mapping T on [0, 1] by Tx¼ x 2; if x6¼ 1; 1þ k 2þ k; if x¼ 1: 8 > < > :
The mapping T satisfies the condition ðCkÞ but it fails the conditionðCk1Þ, whenever 0\k1\k. Moreover, T satisfies
the conditionðElÞ for l ¼2þk2 .
Throughout, this paper we work in the setting of hyperbolic spaces introduced by Kohlenbach [15].
A hyperbolic space (X, d, W) is a metric space (X, d) together with a convexity mapping W : X2 ½0; 1 ! X satisfying
ðW1Þ dðu; Wðx; y; aÞÞ adðu; xÞ þ ð1 aÞdðu; yÞ; ðW2Þ dðWðx; y; aÞ; Wðx; y; bÞÞ ¼ ja bjdðx; yÞ; ðW3Þ Wðx; y; aÞ ¼ Wðy; x; 1 aÞ;
ðW4Þ dðWðx;z;aÞ;Wðy;w;aÞÞð1aÞdðx;yÞþadðz;wÞ; for all x; y; z; w2 X and a; b 2 ½0; 1.
A metric space is said to be a convex metric space in the sense of Takahashi [25], where a triple (X, d, W) satisfy only ðW1Þ: The concept of hyperbolic spaces in [15] is more restrictive than the hyperbolic type introduced by Goebel and Kirk [5] since ðW1Þ andðW2Þ together are equivalent to (X, d, W) being a space of hyperbolic type in [5]. But it is slightly more general than the hyperbolic space defined in Reich and Shafrir [20] (see [15]). This class of metric spaces in [15] covers all normed linear spaces,R-trees in the sense of Tits, the Hilbert ball with the hyperbolic metric (see [6]), Cartesian products of Hilbert balls, Hadamard manifolds (see [20]), and CAT(0) spaces in the sense of Gromov (see [4]). A thorough discussion of hyperbolic spaces and a detailed treatment of examples can be found in [15] (see also [5, 6, 20]).
If x; y2 X and k 2 ½0; 1; then we use the notation ð1 kÞx ky for Wðx; y; kÞ. The following holds even for the more general setting of convex metric space [25]: for all x; y2 X and k 2 ½0; 1;
dðx; ð1 kÞx kyÞ ¼ kdðx; yÞ and dðy; ð1 kÞx kyÞ ¼ ð1 kÞdðx; yÞ: As consequence,
1x 0y ¼ x; 0x 1y ¼ y and
ð1 kÞx kx ¼ kx ð1 kÞx ¼ x:
A hyperbolic space (X, d, W) is uniformly convex [16] if for any r [ 0 and e2 ð0; 2; there exists d 2 ð0; 1 such that for all a; x; y2 X;
d 1 2x 1 2y; a ð1 dÞr:
provided dðx; aÞ r; dðy; aÞ r and dðx; yÞ er:
A mapping g :ð0; 1Þ ð0; 2 ! ð0; 1; which provid-ing such a d¼ gðr; eÞ for given r [ 0 and e 2 ð0; 2; is called as a modulus of uniform convexity. We call the function g is monotone if it decreases with r (for fixed e), that is gðr2;eÞ gðr1;eÞ; 8r2 r1[ 0:
In [16], Leus¸tean proved that CAT(0) spaces are uni-formly convex hyperbolic spaces with modulus of uniform convexity gðr; eÞ ¼e2
8 quadratic in e: Thus, the class of uniformly convex hyperbolic spaces are a natural gener-alization of both uniformly convex Banach spaces and CAT(0) spaces.
Now, we give the concept of D-convergence and some of its basic properties.
Let C be a nonempty subset of metric space (X, d) and fxng be any bounded sequence in X while diam(C) denote the diameter of C. Consider a continuous functional rað; fxngÞ : X ! Rþ defined by
raðx; fxngÞ ¼ lim sup n!1
dðxn; xÞ; x2 X:
The infimum of rað; fxngÞ over C is said to be the asymptotic radius offxng with respect to C and is denoted by raðC; fxngÞ.
A point z2 C is said to be an asymptotic center of the sequencefxng with respect to C if
raðz; fxngÞ ¼ inffraðx; fxngÞ : x 2 Cg;
the set of all asymptotic centers offxng with respect to C is denoted by ACðC; fxngÞ. This set may be empty, a sin-gleton, or certain infinitely many points.
If the asymptotic radius and the asymptotic center are taken with respect to X, then these are simply denoted by raðX; fxngÞ ¼ raðfxngÞ and ACðX; fxngÞ ¼ ACðfxngÞ; respectively. We know that for x2 X, raðx; fxngÞ ¼ 0 if and only if limn!1xn ¼ x. It is known that every bounded sequence has a unique asymptotic center with respect to each closed convex subset in uniformly convex Banach spaces and even CAT(0) spaces.
The following Lemma is due to Leus¸tean [17] and ensures that this property also holds in a complete uni-formly convex hyperbolic space.
Lemma 2.1 [17, Proposition 3.3] Let (X, d, W) be a complete uniformly convex hyperbolic space with mono-tone modulus of uniform convexityg. Then every bounded sequence fxng in X has a unique asymptotic center with respect to any nonempty closed convex subset C of X.
Recall that, a sequencefx g in X is said to D-converge
every subsequencefung of fxng. In this case, we write D-limnxn¼ x and call x the D-limit of fxng.
Lemma 2.2 [13] Let (X, d, W) be a uniformly convex hyperbolic space with monotone modulus of uniform con-vexity g. Let x2 X and ftng be a sequence in [a, b] for some a; b2 ð0; 1Þ. If fxng and fyng are sequences in X such that lim sup n!1 dðxn; xÞ c; lim sup n!1 dðyn; xÞ c; lim n!1dðWðxn; yn; tnÞ; xÞ ¼ c;
for some c 0; then limn!1dðxn; ynÞ ¼ 0:
Lemma 2.3 Let (X, d) be complete uniformly convex hyperbolic space with monotone modulus of convexityg, C be a nonempty closed convex subset of X and T : C! C be a mapping which satisfies conditionsðCkÞ (for some k 2 ð0; 1Þ) and (E) on C. Supposefxng is bounded sequence in C such that
lim
n!1dðxn; TxnÞ ¼ 0; then T has a fixed point.
Proof Since fxng is bounded sequence in X, then by Lemma 2.1, has unique asymptotic center in C, i.e.,
ACðC; fxngÞ ¼ fxg is singleton and
limn!1dðxn; TxnÞ ¼ 0. Since T satisfies the condition ðElÞ on C, there exists l [ 1 such that
dðxn; TxÞ ldðxn; TxnÞ þ dðxn; xÞ:
Taking lim sup as n! 1 both the sides, we have raðTx; fxngÞ ¼ lim sup n!1 dðxn; TxÞ lim sup n!1 ½ldðxn; TxnÞ þ dðxn; xÞ lim sup n!1 dðxn ; xÞ ¼ raðx; fxngÞ:
By using the uniqueness of asymptotic center, Tx¼ x, so x is fixed point of T. Hence, F(T) is nonempty. h
Main results
We begin with the definition of Feje´r monotone sequences: Definition 3.1 Let C be a nonempty subset of hyperbolic space X andfxng be a sequence in X. Then fxng is Feje´r monotone with respect to C if for all x2 C and n 2 N,
dðxnþ1; xÞ dðxn; xÞ:
Example 3.1 Let C be a nonempty subset of X, and T : C! C be a quasi-nonexpansive (in particular, nonexpan-sive) mapping such that FðTÞ 6¼ £ and x02 C. Then the sequence fx g of Picard iterates is Feje´r monotone with
We can easily prove the following proposition. Proposition 3.1 Letfxng be a sequence in X and C be a nonempty subset of X. Suppose thatfxng is Feje´r monotone with respect to C, then we have the followings:
(1) fxng is bounded.
(2) The sequencefdðxn; pÞg is decreasing and converges for all p2 FðTÞ.
We now define Picard Normal S-iteration process (PNS) in hyperbolic spaces:
ðPNSÞ Picard normal S-iteration process: Let C be a nonempty closed convex subset of a hyperbolic space X and T : C! C be a mapping which satisfies the condi-tionðCkÞ for some k 2 ð0; 1Þ. For any x12 C the sequence fxng is defined by xnþ1¼ WðTyn; 0; 0Þ yn ¼ Wðzn; Tzn;anÞ zn¼ Wðxn; Txn;bnÞ; n2 N; 8 > > < > > : ð3:1Þ
where fang and fbng are in ½; 1 for all n 2 N and some 2 ð0; 1Þ.
Lemma 3.1 Let C be a nonempty closed convex subset of a hyperbolic space X and T: C! C be a mapping which satisfies the conditionðCkÞ for some k 2 ð0; 1Þ. If fxng is a sequence defined by (3.1), then fxng is Feje´r monotone with respect to F(T).
Proof Since T satisfies the conditionðCkÞ for some k 2 ð0; 1Þ and p 2 FðTÞ; we have
kdðp; TpÞ ¼ 0 dðp; znÞ kdðp; TpÞ ¼ 0 dðp; ynÞ and
kdðp; TpÞ ¼ 0 dðp; xnÞ; for all n2 N so that, we have
dðTp; TznÞ dðp; znÞ dðTp; TynÞ dðp; ynÞ and dðTp; TxnÞ dðp; xnÞ: Using (3.1), we have dðzn; pÞ ¼ dðWðxn; Txn;bnÞ; pÞ ¼ dðð1 bnÞxn bnTxn; pÞ ð1 bnÞdðxn; pÞ þ bndðTxn; pÞ dðxn; pÞ: ð3:2Þ
From (3.1) and (3.2), we have dðyn; pÞ ¼ dðWðzn; Tzn;anÞ; pÞ ¼ dðð1 anÞzn anTzn; pÞ ð1 anÞdðzn; pÞ þ andðTzn; pÞ ð1 anÞdðzn; pÞ þ andðzn; pÞ dðzn; pÞ dðxn; pÞ: ð3:3Þ
Again, using (3.2) and (3.3), we have dðxnþ1; pÞ ¼ dðWðTyn; 0; 0Þ; pÞ
¼ dðTyn; pÞ dðyn; pÞ
dðxn; pÞ; ð3:4Þ
that is, dðxnþ1; pÞ dðxn; pÞ for all p 2 FðTÞ: Thus, fxng is Feje´r monotone with respect to F(T). h Lemma 3.2 Let C be a nonempty closed convex subset of a complete uniformly convex hyperbolic space with monotone modulus of uniform convexityg and T : C! C be a mapping which satisfies the condition ðCkÞ for some k2 ð0; 1Þ. If fxng is a sequence defined by (3.1), then F(T) is nonempty if and only if the sequencefxng is bounded and limn!1dðxn; TxnÞ ¼ 0.
Proof Suppose that the fixed point set F(T) is nonempty and p2 FðTÞ: Then by Lemma3.1,fxng is Feje´r monotone with respect to F(T) and hence by Proposition3.1,fxng is bounded and limn!1dðxn; pÞ exists, let limn!1dðxn; pÞ ¼ c 0:
(i) If c¼ 0, we obviously have dðxn; TxnÞ dðxn; pÞ þ dðTxn; pÞ
2dðxn; pÞ;
by taking lim as n! 1 on both the sides above inequality, we have
lim
n!1dðxn; TxnÞ ¼ 0:
(ii) If c [ 0, since T satisfies the condition ðCkÞ for some k2 ð0; 1Þ and p 2 FðTÞ; we have
dðTxn; pÞ dðxn; pÞ;
taking lim sup as n! 1 both the sides, we get lim sup
n!1 dðTxn; pÞ c:
Taking lim sup as n! 1 both the sides in (3.2), we have
lim sup n!1
Since
dðxnþ1; pÞ dðzn; pÞ;
therefore, we take lim inf as n! 1 both the sides, we get lim inf
n!1 dðxnþ1; pÞ lim infn!1 dðzn; pÞ c lim inf
n!1 dðzn; pÞ
ð3:6Þ
From (3.5) and (3.6), we have lim n!1dðzn; pÞ ¼ c; it implies that c¼ lim sup n!1 dðzn; pÞ ¼ lim sup n!1 ½dðWðxn; Txn;bnÞ; pÞ ¼ lim sup n!1 ½dðð1 bnÞxn bn Txn; pÞ lim sup n!1 ½ð1 bnÞdðxn; pÞ þ bndðTxn; pÞ ð1 bnÞ lim sup n!1 dðxn; pÞ þ bnlim sup n!1 dðTxn; pÞ ¼ c: Hence, it follows from Lemma2.2, we have
lim
n!1dðxn; TxnÞ ¼0:
Conversely, suppose that sequence fxng is bounded and limn!1dðxn; TxnÞ ¼ 0: Hence, it holds all the assump-tion of Lemma 2.3, so we have Tx¼ x, i.e., F(T) is
nonempty. h
Theorem 3.1 Let C be a nonempty closed convex subset of a complete uniformly convex hyperbolic space X with monotone modulus of uniform convexityg and T : C! C be a mapping which satisfies conditions ðCkÞ (for some k2 ð0; 1Þ) and (E) on C with FðTÞ 6¼ £. If fxng is the sequence defined by (3.1), then the sequence fxng D-con-verges to a fixed point of T.
Proof From Lemma 3.2, we observe that fxng is a bounded sequence therefore, fxng has a D-convergent subsequence. We now prove that every D-convergent subsequence offxng has unique D-limit F(T). For this, let u and v D-limits of the subsequencesfung and fvng of fxng respectively. By Lemma 2.1, ACðC; fungÞ ¼ fug and ACðC; fvngÞ ¼ fvg. By Lemma 3.2, we have limn!1dðun; TunÞ ¼ 0.
We claim that u and v are fixed points of T and it is unique.
By Lemma2.3, u and v are fixed points of T. Now we show that u¼ v. If not, then by uniqueness of asymptotic center lim sup n!1 dðxn ; uÞ ¼ lim sup n!1 dðun ; uÞ \ lim sup n!1 dðun; vÞ ¼ lim sup n!1 dðxn; vÞ ¼ lim sup n!1 dðvn ; vÞ \ lim sup n!1 dðvn; uÞ ¼ lim sup n!1 dðxn; uÞ;
which is a contradiction. Hence u¼ v, the sequence fxng D-converges to a fixed point of T. h Theorem 3.2 Let C be a nonempty closed convex subset of a complete uniformly convex hyperbolic space X with monotone modulus of uniform convexityg and T : C! C be a mapping which satisfies conditions ðCkÞ (for some k 2 ð0; 1ÞÞ and (E) on C with FðTÞ 6¼ £. Then the sequence fxng which is defined by (3.1), converges strongly to some fixed point of T if and only if lim infn!1Dðxn; FðTÞÞ ¼ 0, where Dðxn; FðTÞÞ ¼ infx2FðTÞdðxn; xÞ.
Proof Necessity is obvious, we have to prove only suf-ficient part. First, we show that the fixed point set F(T) is closed, letfxng be a sequence in F(T) which converges to some point z2 C. As
kdðxn; TxnÞ ¼ 0 dðxn; zÞ;
in view of the condition ðCkÞ, we have dðxn; TzÞ ¼ dðTxn; TzÞ dðxn; zÞ:
By taking the limit of both sides we obtain lim
n!1dðxn; TzÞ limn!1dðxn; zÞ ¼ 0:
In view of the uniqueness of the limit, we have z¼ Tz, so that F(T) is closed. Suppose
lim inf
n!1 Dðxn; FðTÞÞ ¼ 0: From (3.4)
Dðxnþ1; FðTÞÞ Dðxn; FðTÞÞ;
it follows from Lemma 3.1 and Proposition 3.1 that limn!1dðxn; FðTÞÞ exists. Hence we know that limn!1Dðxn; FðTÞÞ ¼ 0.
Consider a subsequence fxnkg of fxng such that
dðxnk; pkÞ\
1 2k;
dðxnkþ1; pkÞ dðxnk; pkÞ\
1 2k; which implies that
dðpkþ1; pkÞ dðpkþ1; xnkþ1Þ þ dðxnkþ1; pkÞ \ 1 2kþ1þ 1 2k \ 1 2k1:
This shows thatfpkg is a Cauchy sequence. Since F(T) is closed,fpkg is a convergent sequence. Let limk!1pk¼ p. Then we know thatfxng converges to p. In fact, since
dðxnk; pÞ dðxnk; pkÞ þ dðpk; pÞ ! 0; ask ! 1;
we have limk!1dðxnk; pÞ ¼ 0. Since limn!1dðxn; pÞ
exists, the sequencefxng is convergent to p.
We recall the definition of condition (I) due to Senter and Doston [22], define as follows:
Definition 3.2 [22] Let C be a nonempty subset of a metric space X. A mapping T : C! C with nonempty fixed point set F(T) in C is said to satisfy Condition (I) if there is a nondecreasing function f :½0; 1Þ ! ½0; 1Þ with fð0Þ ¼ 0; f ðtÞ [ 0 for all t2 ð0; 1Þ, such that dðx; TxÞ f ðDðx; FðTÞÞÞ for all x2 C, where Dðx; FðTÞÞÞ ¼ inffdðx; pÞ : p 2 FðTÞg.
Theorem 3.3 Let C be a nonempty closed convex subset of a complete uniformly convex hyperbolic space X with monotone modulus of uniform convexityg and T : C! C be a mapping which satisfies conditions ðCkÞ (for some k2 ð0; 1Þ) and (E) on C. Moreover, T satisfies the condi-tion (I) with FðTÞ 6¼ £. If fxng is the sequence defined by (3.1), then the sequence fxng converges strongly to some fixed point of T.
Proof As in proof of Theorem3.2, it can be shown that F(T) is closed. Observe that by Lemma 3.1, we have limn!1dðxn; TxnÞ ¼ 0. It follows from the condition (I) that lim n!1fðDðxn; FðTÞÞ limn!1dðxn; TxnÞ ¼ 0: Therefore, we have lim n!1fðDðxn; FðTÞÞÞ ¼ 0:
Since f :½0; 1 ! ½0; 1Þ is a nondecreasing mapping satisfying fð0Þ ¼ 0 and f ðtÞ [ 0 for all t 2 ð0; 1Þ, we have limn!1dðxn; FðTÞÞ ¼ 0. Rest of the proof follows in lines
of Theorem3.2. h
In the view of the Remark1.1the following Corollaries are trivially true.
Corollary 3.1 Let C be a nonempty closed convex subset of a complete uniformly convex hyperbolic space X with monotone modulus of uniform convexityg and T : C! C be a mapping which satisfies conditions ðCkÞ (for some k2 ð0; 1Þ) and (E) on C with FðTÞ 6¼ £. If fxng is the sequence defined by (for each x12 C)
xnþ1 ¼ WðTyn; 0; 0Þ
yn¼ Wðxn; Txn;anÞ; n2 N;
ð3:7Þ then the sequence fxng D-converges to a fixed point of T. Corollary 3.2 Under the assumption of Corollary 3.1
with FðTÞ 6¼ £. The sequence fxng which is defined by (3.7), converges strongly to some fixed point of T if and only if limn!1inf Dðxn; FðTÞÞ ¼ 0; where Dðxn; FðTÞÞ ¼ infx2FðTÞdðxn; xÞ:
Corollary 3.3 Under the assumption of Corollary 3.1
with FðTÞ 6¼ £ and T satisfies the condition (I). The sequencefxng which is defined by (3.7), converges strongly to some fixed point of T.
In the view of the Remark 2.2, we have the following Corollaries:
Corollary 3.4 Let C be a nonempty closed convex subset of a complete uniformly convex hyperbolic space X with monotone modulus of uniform convexityg and T : C! C be a SKC mapping with FðTÞ 6¼ £. The sequence fxng defined by (3.1), D-converges to a fixed point of T. Corollary 3.5 Under the assumption of Corollary 3.4
with FðTÞ 6¼ £. The sequence fxng which is defined by (3.1), converges strongly to some fixed point of T if and only if lim infn!1Dðxn; FðTÞÞ ¼ 0, where Dðxn; FðTÞÞ ¼ infx2FðTÞdðxn; xÞ:
Corollary 3.6 Under the assumption of Corollary 3.4
with FðTÞ 6¼ £ and T satisfies the condition (I). The sequencefxng which is defined by (3.1), converges strongly to some fixed point of T.
Corollary 3.7 Let C be a nonempty closed convex subset of a complete uniformly convex hyperbolic space X with monotone modulus of uniform convexityg and T : C! C be a SKC mapping with FðTÞ 6¼ £. Then the sequence fxng defined by (3.7), D-converges to a fixed point of T. Corollary 3.8 Under the assumption of Corollary 3.7
with FðTÞ 6¼ £. The sequence fxng which is defined by (3.1), converges strongly to some fixed point of T if and only if lim infn!1Dðxn; FðTÞÞ ¼ 0; where Dðxn; FðTÞÞ ¼ infx2FðTÞdðxn; xÞ:
Corollary 3.9 Under the assumption of Corollary 3.7
sequencefxng which is defined by (3.1), converges strongly to some fixed point of T.
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